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## Teacher Goals and Dilemmas in the Use of Mathematical Representations

The purpose of this article is to describe the reported goals of teachers participating in professional development related to their efforts to consider multiple representations in their teaching. Through analysis of monthly written reflections and group discussion, we describe 3 teacher dilemmas that emerged related to their efforts to consider multiple representations in their teaching: (a) equitable practices when particular representations result in differential success, (b) a teacher's need to balance exposure and choice, and (c) a potential dilemma related to conflicts between competing goals. We provide suggestions of how mathematics educators might use these findings to support future professional development efforts.

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## Teaching Structure in Algebra

Exercises about glue and trees and addresses can inoculate students against their notorious tendency to reduce incorrectly when simplifying expressions.

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## Exploring Function Transformations Using the Common Core

Tying your teaching approach to the Common Core Standard for Geometry and Congruence will help students understand why functions behave as they do.

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## Norms and Mathematical Proficiency

How do classroom behavioral expectations support the development of students' mathematical reasoning? A sixth-grade teacher and his students developed this example while discussing a ratio comparison problem.

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## The healthy vending machine

A good problem can capture students' curiosity and can serve many functions in the elementary school classroom: to introduce specific concepts the teacher can build on once students recognize the need for additional mathematics or to help students see where to apply already-learned concepts. We encourage teachers to use the monthly problem in their own classrooms and report solutions, strategies, reflections, and misconceptions to the journal audience.

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## Is That Square Really a Circle?

Considering circles in taxicab geometry helps students with Euclidean concepts.

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## Core Conversations with Educative Dragging

The Measure-Trace-Algebratize (MTA) approach allows students to uncover algebraic relationships within familiar geometric objects.

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## Tapering Timbers: Finding the Volume of Conical Frustums

Many everyday objects–paper cups, muffins, and flowerpots–are examples of conical frustums. Shouldn't the volume of such figures have a central place in the geometry curriculum?

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## Enriching Number Knowledge

Exploring number systems of other cultures helps students deepen mental computation fluency, knowledge of place value, and equivalent representations for numbers.

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## Sift like Eratosthenes

The concrete, pictorial, and abstract methods of this lesson give students access to investigate, isolate, define, and use prime numbers.